How Quantum Physics Converts Momentum and Position into Probabilities

Quantum physics, unlike classical physics, is completely nondeterministic. You can never know the precise position and momentum of a particle at any one time. You can give only probabilities for these linked measurements.

In quantum physics, the state of a particle is described by a wave function,

The wave function describes the de Broglie wave of a particle, giving its amplitude as a function of position and time. (The de Broglie wave is named after the physicist Louis de Broglie.)

Note that the wave function gives a particle's amplitude, not intensity; if you want to find the intensity of the wave function, you have to square it:

The intensity of a wave is what's equal to the probability that the particle will be at that position at that time.

That's how quantum physics converts issues of momentum and position into probabilities: by using a wave function, whose square tells you the probability density that a particle will occupy a particular position or have a particular momentum. In other words,

is the probability that the particle will be found in the volume element d3r, located at position r at time t.

Besides the position-space wave function

there's also a momentum-space version of the wave function:

In quantum physics, you study many aspects of the wave function — the wave functions of free particles, the wave functions of particles trapped inside potentials, of identical particles hitting each other, of particles in harmonic oscillation, of light scattering from particles, and more. Using this kind of physics, you can predict the probabilistic behavior of all kinds of physical systems.